We will investigate the scenario in which the Standard Model (SM) Higgs sector and its two-doublet extension (called the Two Higgs Doublet Model or 2HDM) are the “portal” for the interactions between the Standard Model and a fermionic Dark Matter (DM) candidate. The latter is the lightest stable neutral particle of a family of vector-like leptons (VLLs). We will provide an extensive overview of this scenario combining the constraints coming purely from DM phenomenology with more general constraints like Electroweak Precision Test (EWPT) as well as with collider searches. In the case that the new fermionic sector interacts with the SM Higgs sector, constraints from DM phenomenology force the new states to lie above the TeV scale. This requirement is relaxed in the case of 2HDM. Nevertheless, strong constraints coming from EWPTs and the Renormalization Group Equations (RGEs) limit the impact of VLFs on collider phenomenology.

Eur. Phys. J. C
Dark matter phenomenology of SM and enlarged Higgs sectors extended with vector-like leptons
Andrei Angelescu 1
Giorgio Arcadi 0
0 Max Planck Institüt für Kernphysik , Saupfercheckweg 1, 69117 Heidelberg , Germany
1 Laboratoire de Physique Théorique, Université Paris-Saclay , CNRS, 91405 Orsay , France
We will investigate the scenario in which the Standard Model (SM) Higgs sector and its two-doublet extension (called the Two Higgs Doublet Model or 2HDM) are the “portal” for the interactions between the Standard Model and a fermionic Dark Matter (DM) candidate. The latter is the lightest stable neutral particle of a family of vectorlike leptons (VLLs). We will provide an extensive overview of this scenario combining the constraints coming purely from DM phenomenology with more general constraints like Electroweak Precision Test (EWPT) as well as with collider searches. In the case that the new fermionic sector interacts with the SM Higgs sector, constraints from DM phenomenology force the new states to lie above the TeV scale. This requirement is relaxed in the case of 2HDM. Nevertheless, strong constraints coming from EWPTs and the Renormalization Group Equations (RGEs) limit the impact of VLFs on collider phenomenology.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 Vector-like extensions of the Standard Model . . . . .
3 Two Higgs doublet models . . . . . . . . . . . . . . .
4 Conclusions . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Weakly Interacting Massive Particles (WIMPs) represent
probably the most popular class of Dark Matter (DM)
candidates. Among the features which make this kind of candidates
so attractive, it is for sure worth mentioning the production
mechanism. WIMP DM was indeed part of the primordial
thermal bath at early stages of the history of the Universe
and decoupled (freeze-out) at later stages, when the
temperature was below their mass (i.e. non-relativistic decoupling),
since the interactions with the SM particles were not
efficient anymore with respect to the Hubble expansion rates.
Under the assumption of standard cosmological history, the
comoving abundance of the DM is set by a single particle
physics input, namely the thermally averaged pair
annihilation cross section. The experimentally favored value of
DM abundance, expressed by the quantity h2 ≈ 0.12 [
1
],
corresponds to a thermally averaged cross section σ v ∼
10−26 cm3 s−1. Interactions of this size are potentially
accessible to a broad variety of search strategies, ranging from
direct/indirect detection to production at colliders, making
the WIMP paradigm highly testable.
From the point of view of model building, WIMP
frameworks feature interactions between pairs of DM particles (in
order to guarantee the cosmological stability of the DM,
operators with a single DM field are in general forbidden, e.g.
through a symmetry) and pair of SM states, induced by
suitable mediator fields. The simplest option, in this sense, is
probably represented by s-channel electrically neutral
mediators, dubbed “portals”, which can couple the DM with SM
fermions (see e.g. [
2–4
]), although couplings with the SM
gauge bosons might also be feasible [
5–8
]. The DM relic
density is thus determined via s-channel exchange of the
mediator states. By simple crossing symmetry arguments these
processes can be, for example, related to the rate of DM Direct
Detection, induced by the t-channel interaction between the
DM and the SM quarks, and to the ones of DM pair
production at colliders, which can be probed mostly through
mono-jet events [
9–12
].
Interestingly, the SM features two potential s-channel
mediators, namely the Z and the Higgs bosons. One possible
result concerns “Z-portal” DM [
13
] scenarios. However, they
are rather contrived, since, because of gauge invariance,
interactions between a SM singlet DM and the Z can arise only at
the non-renormalizable level [
14,15
]. “Higgs portal” models
are instead very popular, although rather constrained [
16–
20
], since a DM spin-0 (1), even if it is a singlet with respect
to the SM gauge group, can interact with the SM Higgs
doublet H via four-field operators connecting the bilinear H H †
with a DM pair and giving rise, after electroweak (EW)
symmetry breaking, to an effective vertex between a DM pair and
the physical Higgs field h.
The fermionic “Higgs portal” is instead a dimension-5
operator. Furthermore this is strongly constrained, also with
respect to the scalar and vector DM cases, because of the
strong direct detection rates accompanied by a velocity
suppressed annihilation cross section [
18,19
].
In order to couple at the renormalizable level with the
Z and/or Higgs bosons, the fermionic DM should feature a
(small) hyper- and SU (2) charged component. This could be
realized through the mixing of a pure SM singlet and extra
states with non-trivial quantum numbers under SU (2)×U (1)
(see e.g. [
21–24
] for some constructions). The DM should
then be a stable neutral state belonging to a new, non-trivial
particle sector.
New chiral fermions, with mass originating from EWSB,
are strongly disfavored experimentally [
25
]. More suitable
options are instead represented by fermions belonging to a
real representation or forming vector-like pairs.
In this work we will consider this last option and then
extend the fermionic content of the SM with a “family”
of new fields, with analogous quantum numbers as the SM
leptons and the right-handed neutrinos, and with bare mass
terms, which are allowed by gauge symmetry, since the new
fermions are vector-like under the SM gauge group.
Therefore, these fields are dubbed “vector-like leptons” (VLLs).
In the absence of mixing with SM leptons, the lightest new
fermionic state, if electrically neutral, constitutes a DM
candidate. In this setup the DM is coupled, through Yukawa
interactions, with the SM Higgs and with the Z and W bosons,
featuring, in general, non-zero components charged under
hypercharge and weak isospin.
This kind of scenario is, unfortunately, very strongly
constrained since the Higgs and Z-boson mediate Spin
Independent (SI) interactions between the DM and the
nucleons, which are in increasing tension with experimental
constraints. Similarly to the Higgs and Z-portal models it is
possible to comply with these limits and achieve, at the same time,
the correct relic density only for rather heavy DM masses or,
possibly, in the presence of coannihilation processes, thus
implying mass degeneracies in the new fermionic sector.
A more interesting option would consist in enlarging the
mediator sector by considering two Higgs doublets (2HDM).
Besides the still rather fine-tuned possibilities of s-channel
resonances and coannihilations, it is possible, in this scenario,
to enhance the DM annihilation cross section, marginally
affecting its scattering rate on nucleons, through annihilation
into extra Higgs bosons, especially the charged ones, as final
states, provided that the latter are light enough. This last
possibility evidences an interesting complementarity with
collider searches of extra Higgs bosons. Lower limits on their
masses would automatically constrain the range of viable
DM masses.
LHC searches of new scalar states can be themselves
influenced by the presence of new vector-like fermions since
electrically charged VL fermions (which will be present in the
SU (2) multiplet which the DM belongs to) or even color
charged VL fermions (we will not consider explicitly this
possibility here) can modify di-boson signal rates. For this
reason 2HDM+VLFs models have attracted great attention
in the recent times since they allowed for the interpretation
of the 750 GeV diphoton excess [
26–38
], announced by the
LHC collaboration in December 2015 [
39–42
], but not
confirmed by the 2016 data [
43,44
].
The parameters of the theory are constrained not only by
the DM and collider phenomenology. The size of the
couplings of the new fermions to the 125 GeV Higgs is
constrained by Electro-Weak Precision Tests (EWPT). A further
strong upper bound on these couplings, as well as the ones
with the other Higgs states, comes from the RG running of
the gauge and the quartic couplings of the scalar potential.
In particular, the latter get strong negative contributions
proportional to the fourth power of the Yukawa couplings of the
VLLs, such that the scalar potential might be destabilized
even at collider energy scales, unless new degrees of freedom
are added. An important consequence of this broad variety
of strong constraints is that, as will be shown below, in the
parameter region corresponding to viable DM
phenomenology, the collider pair production of the DM itself, through
decays of the electrically neutral Higgs states, is strongly
disfavored.
This work aims at an extensive overview of the
phenomenology of the SM and several realizations of 2HDM
extended with a sector of vector-like fermions with a stable
neutral particle as lightest stable state and DM candidate.
The paper is organized as follows. We will firstly
introduce, at the beginning of Sect. 2, the “family” of vector-like
fermions. The remainder of the section will be dedicated
to a brief overview of the SM+VLLs scenario. Firstly, we
will briefly illustrate the general constraints coming from
the modification of the Higgs signal strengths and the
Electroweak Precision Tests (EWPT), and afterwards focus on
the DM phenomenology. Along similar lines, an analysis
for the 2HDM will then be performed in Sect. 3. After a
short review of the general aspects of 2HDMs, we will
perform a more detailed analysis of the constraints from EWPT
and Higgs signal strengths and add to them the RGE
constraints. After the analysis of the DM phenomenology, we
will briefly discuss the limits/prospects, for our scenario, of
collider searches. Finally, we will summarize our results in
Sect. 3.7 and conclude in Sect. 4.
2 Vector-like extensions of the Standard Model
In this section we will review how introducing vector-like
leptons affects the SM Higgs sector. As already pointed out,
the impact is mostly twofold. First of all, they generate
additional loop contributions to the couplings of the Higgs boson
to two photons, giving rise to deviations of the corresponding
signal strength with respect to the SM prediction. In
addition, the presence of vector-like leptons is typically
associated with sensitive departures from experimental limits for
the EW precision observables. In order to have viable values
of the Higgs signal strengths and precision observables, one
should impose definite relations for the Yukawa couplings
and masses of the new VLLs. The same relations will hold,
up to slight modifications, also in the 2HDM case.
2.1 The vector-like “family”
In this work we will assume that the SM and, afterwards,
the 2HDM Higgs sectors can be extended by “families” of
vector-like fermions (VLFs). By a family we understand a set
of two SU (2)L singlets and one SU (2)L doublet, belonging
to a SU (3)c representation Rc, and with their hypercharge
determined by a single parameter, Y . For the moment, we
will keep the discussion general and later on specialize on
possible DM candidates. The new fields can be schematically
labeled
DL,R ∼ (Rc, 2, Y − 1/2), UL,R ∼ (Rc, 1, Y ),
DL,R ∼ (Rc, 1, Y − 1),
(1)
so that the couplings to the SM Higgs doublet, H
=
0 v√+2h T , are parametrized by the following Lagrangian:
−LVLF = yUR DL H˜ UR + yUL UL H˜ †DR
+y DR DL H DR + y DL DL H †DR
+MU DDL DR + MU UL UR + MD DL DR + h.c.,
where we have considered the following decomposition for
the SU (2) doublets: DL,R ≡ U D TL,R .
For simplicity we will assume that all the couplings are
real and that the mixing between the VLFs and the SM
fermions is negligible. Later on, when specializing on DM
phenomenology, we will forbid the SM fermion–VL fermion
mixing through a global Z2 symmetry.
After electroweak symmetry breaking (EWSB), there is a
mixing in the “up” (U , U ) and “down” (D , D) sectors. The
“up” VL fermions have charge QU = Y , while the “down”
fermions have charge Q D = (Y − 1). The mass matrices in
the two sectors are
MU =
MD =
MU
yUR v/√2
MD
y DR v/√2
yUL v/√2
MU D
y DL v/√2
MU D
,
,
,
with v = 246 GeV, and they are bi-diagonalized as follows:
ULF · MF · URF † =
m F1 0
0 m F2
ULF =
cF s F
−sLF cLF , URF =
L L
cF s F
−sRF cRRF ,
R
where the sub/superscripts F = U, D distinguish between
the two sectors and cLF/R = cos θLF/R , sLF/R = sin θLF/R .
Throughout this work we will denote the lighter mass
eigenstate as F1. The limit where one of the singlets is decoupled,
e.g. when yUR = yUL = 0 and MU → ∞, has already
been studied in detail in Ref. [
45
]. As we will see below the
mixing structure in Eq. 3 is strongly constrained by the
electroweak precision tests (EWPT) and by the Higgs couplings
measurements.
2.2 Electroweak precision tests
Extending the SM with vector-like fermions leads, in general,
to the deviation of the Electroweak precision observables
S and T from their respective experimental limits.
Assuming negligible mixing between the SM and the vector-like
fermions, the limits on S and T can be directly translated
into limits on the Yukawa couplings and masses of the new
fermions; in the limit in which the former go to zero, no
constraints from EWPT apply.
Sizable values of the Yukawa couplings of the VLFs can
nevertheless be obtained while still complying with the limits
on the T parameter by relying (at least approximately) on a
custodial limit:
(2)
MD = MU ,
yUL = y DL ,
yUR = y DR ,
(5)
(3)
(4)
which is equivalent to imposing equal mass matrices in the
isospin-up and isospin-down sectors. Clearly, the custodial
limit can be achieved only by considering “full families”
of VLFs, i.e. a corresponding SU(2) singlet for each of the
components of the doublet, as done in this work.
On the contrary, there is no symmetry protecting the S
parameter, which means that, in some cases, it will impose
more relevant constraints than the T parameter. The
constraints on S can be nevertheless partially relaxed by taking
advantage of the correlation among the S and T parameters,
illustrated in Fig. 1, by allowing for a small deviation from
the custodial limit, i.e. T 0.
2.3 Higgs couplings
We now turn to the second constraint coming from the
Higgs couplings measurements. In the presence of
vectorlike fermions, its couplings to gauge bosons receive
additional contributions, originating from triangle loops in which
the new fermions are exchanged. No new decay channels into
VLFs are instead present since, because of constraints from
direct searches at colliders, the VLFs should be heavier than
the SM Higgs.
The SM Higgs loop-induced partial decay widths into
massless gauge bosons, hVV, V = g, γ , can be
schematically expressed as hVV ∝ |AShMVV + AVhVLVF|2, where AShMVV
and AVhVLVF represent the amplitudes associated, respectively,
with the SM and VLF contributions. Throughout this work
we will only consider the case of a family of color-neutral
VLFs ( Rc = 1); as a consequence the new physics sector
influences mostly hγ γ and therefore the h → γ γ signal
strength, μγ γ .1 The corresponding amplitude is given by
hγ γ
AVLF =
F=U,D
i=1,2
Q2F v(CF )ii A1h/2(τFh ),
m Fi i
where τFhi = 4mm2h2F , while A1h/2 is a loop form factor whose
i
definition is given e.g. in [
49
]. The matrix CF is defined as
CF = UFL · YF · (UFR )†,
1
YF = ∂v MF = √2
0
y FR
h
y FL
h
0
(6)
1 Note that μh Zγ is also affected by the VLFs, but the uncertainties on
this signal strength are too large to constrain the extended fermionic
sector [
47,48
].
(8)
(9)
(10)
2 Alternatively one could think about a cancellation between the
contributions of the “up” and “down” sectors. In order to have a DM candidate
we will consider, however, in this work the case that the up sector is
made by electrically neutral states, so that they do not actually
contribute to μγ γ . On general grounds, a cancellation between the up-type
and down-type contribution would be anyway difficult to realize since
it would require a very strong deviation from the custodial symmetry
limit, which is disfavored by EWPT.
3 This constraint on the Yukawa coupling can be relaxed by adding
more families of VLF and/or considering higher values of Y . However,
we shall not consider these cases throughout this work.
For a 125 GeV Higgs we can reliably approximate the loop
function A1h/2(τ ) with its asymptotic value, for A1h/2(0) =
4/3, such that the expression (6) simplifies to
hγ γ
AVLF = A1h/2(0)
−2v2 yhFL y FR
h
F=U,D 2MF MU D − v2 yhFL y FR .
h
Experimental measurements do not exhibit statistically
relevant deviations of μγ γ from the SM prediction [
47, 48
],
which implies essentially two possibilities: AVhγLγL 0 or
AVhγLγL −2AShMγγ . As evident from Eq. (8), the first
possibility is easily realized by setting to zero one of the y FL,R
h
couplings.2 The other is instead more complicated to realize.
Assuming Y = 0 (as will be done for the rest of the paper),
such that only D-type states contribute to μγ γ , and setting
for simplicity MD = MU D and yhDL = −yhDR = yhD , which
implies that the two mass eigenstates will have the same mass
m D , the relation to impose becomes
hγ γ 4
AVLL = 3
yhD v
m D
2
hγ γ
−2ASM
13,
which is impossible to satisfy since yhD v/m D is smaller than
2 (or equal, for MD = MU D = 0).3 Unless stated otherwise,
we will always consider, for both the SM and the 2HDM
cases, an assignation of the Yukawa couplings of the VLFs
hγ γ
such that AVLL = 0.
2.4 DM phenomenology
A DM candidate is introduced, in our setup, by
considering a “family” of vector leptons coupled with the SM Higgs
doublet according to the following lagrangian:
− LV L L = y NR L L H˜ N R + yhNL N L H˜ † L R
h
+ yhE R L L H E R + yhEL E L H † L R
+ ML L L L R + MN N L N R
+ ME E L E R + h.c..
To guarantee the stability of the DM candidate, we impose
a global Z2 symmetry under which the vector-like leptons
are odd and the SM is even (a supersymmetric analogue
is the well-known R-parity). After EW symmetry breaking
a mixing between the vector-like fermions is generated, as
described by the following mass matrices:
MN =
ML =
MN v yhNL
v yhNR ML
ME v yhEL
v yhER ML
,
where v = v/√2 174 GeV. Note that the Z2 symmetry
prevents mixing between the VLLs and the SM fermions. In
order to pass from the interaction to the mass basis one has
to bidiagonalize the above matrices as
L · MN · (URN )† = diag(m N1 , m N2 ),
U N
ULE · ME · (URE )† = diag(m E1 , m E2 ),
with the unitary matrices ULF,R , F = N , E written explicitly
as
ULF,R =
where
tan 2θLN =
tan 2θRN =
cos θLF,R sin θLF,R
− sin θLF,R cos θLF,R
,
2√2v ML yhNL + MN yhNR
2M 2 y NL 2
L − 2M N2 − v2 | h | − |yhNR |2
2√2v MN yhNL + ML yhNR
2M 2 y NL 2
L − 2M N2 + v2 | h | − |yhNR |2
,
The corresponding expressions for θLE,R can be found from
the ones above by replacing MN → ME and yhNL,R → yhEL,R .
The DM candidate N1 (i.e. the lighter VL neutrino) is in
general a mixture of the SU (2) singlet (with null hypercharge)
NL,R and doublet NL,R . As a consequence it is coupled with
the Higgs scalar h as well as with the SM gauge bosons W ±
and Z . These couplings are given by
yh N1 N1 =
cos θNL sin θNR yhNL + cos θNR sin θNL yhNR ,
√2
yV,Z N1 N1 = 4 cogs θW (sin2 θLN + sin2 θRN ),
yA,Z N1 N1 = 4 cogs θW (sin2 θLN − sin2 θRN ),
g
√ (sin θLN sin θLE + sin θRN sin θRE ),
yV,W N1 E1 = 2 2
(17)
(18)
g
√ (sin θLN sin θLE − sin θRN sin θRE ),
yA,W N1 E1 = 2 2
(15)
where, for convenience, we have expressed the couplings
with the Z and W bosons in terms of vectorial and axial
combinations.
The DM relic density can be determined through the
WIMP paradigm as a function of the DM thermally averaged
pair annihilation cross section, formally defined (excluding
coannihilations) as [
50
]:
which is in turn a function of the couplings reported in
Eq. (15). The possible DM annihilation processes consist in
annihilations into SM fermions pairs, induced by s-channel
exchange of the h and Z bosons, and into W +W −, Z Z ,
Z h, and hh, induced also by t-channel exchange of the
neutral states N1,2 (E1,2 for the W +W − final state). In order to
precisely determine the DM relic density we have
numerically computed (16) through the package micromegas [
51
].
We will nevertheless provide some simple approximations
to facilitate the comprehension of the relationship between
the DM relic density and the relevant parameters of the
theory, obtained by the conventional velocity expansion [
52
]
σ v ≈ a + 2b/x (using σ v ≈ a + bv2/3, v2 = 6/x ) and
taking only, if non-vanishing, the leading, s-wave, coefficient
a.4
In the case of annihilation into f¯ f final states, the only
non-vanishing contribution in the v → 0 limit is the one
associated to the s-channel Z -exchange:
m2N1
σ v f f ≈ 8π π((4m2N1 − m2Z )2 + m2Z 2Z )
×
ncf (|V f |2 + | A f |2)|yV,Z N1 N1 |2,
where V f and A f are the vectorial and axial couplings of the
Z -boson and the SM fermions:
g 2
V f = 2cW (−2q f sW + T f3),
A f = 2cgW T f3,
4 As already pointed out, the reader should take these expressions as
illustrative. The contribution of the coefficient b is not necessarily
negligible. We also recall that the velocity expansion is not valid in the
presence of s-channel resonances (relevant in the 2HDM section),
coannihilations and thresholds corresponding to the opening of annihilation
channels [
53
].
(sin θLN )2 + (sin θRN )2
4
Here tW = tan θW . We have
g4
σ v Z Z ≈ 32π cW4 m2Z
The achievement of the correct relic density through DM
annihilations can be potentially in tension with limits from
direct detection experiments. Indeed, DM interactions with
SM quarks, mediated by t-channel exchange of Z and h
bosons, induce both Spin Independent (SI) and Spin
Dependent (SD) scattering processes of the DM with nuclei of target
detectors.
The corresponding cross sections, focusing for simplicity
on the scattering on protons, are given by
σNS1I p,Z =
×
2
μN1 1
π m4Z |yV,Z N1 N1 |
2
Z
1 + A
Vu +
Z
2 − A
Vd
2
5 For simplicity we have assumed that the t-channel diagrams are
dominated by the exchange of the lightest mass eigenstate.
while ncf is the color factor and sW = sin θW and cW =
cos θW . The cross sections of the other relevant final states
can be instead estimated as5:
g4tW
σ v W +W − ≈ 16π m2W ((sin θLN )2 + (sin θRN )2)2
((sin θLN sin θLE )2 + (sin θRN sin θRE )2)2
2
fq + 27 fT G
⎠
q=c,b,t
(22)
⎞
2
,
nu SnA)
1
(SpA + SnA)2
(23)
In order to comply with the stringent limits by the LUX
experiment [
55
] which impose, for DM masses of the order of few
hundreds GeV, a cross section of the order of 10−45 cm2, we
need to require sin2 θLN + sin2 θRN ∼ 10−(1÷2).
We have computed the main DM observables, i.e. relic
density and SI scattering cross section, for a sample of
model points generating by scanning on the parameters
(yhNL,R , yhEL , MN , ME , ML ), while we set yhER = 0 in order
hγ γ
to achieve AN P = 0, over the following range:
yhNL,R ∈ [
10−3, 1
],
yhEL ∈ [5 × 10−3, 3],
MN ∈ [100 GeV, 5 TeV],
ME = ML ∈ [300 GeV, 5 TeV],
with the additional requirement of not exceeding the limits
from EWPT.
The results of our analysis are reported in Fig. 2. The
figure shows the set of points featuring the correct DM relic
density in the bidimensional plane (m N1 , σSI). As evident,
the very strong constraints from the Z -mediated DM
scattering on nucleons rule out the parameter space corresponding
to thermal DM unless its mass is approximately above 2 TeV.
This result is very similar to what is obtained in the generic
scenario dubbed Z -portal [
13
], in which the SM Z boson
mediates the interactions between the SM states and a Dirac
(25)
1.×10−41
1.×10−42
fermion DM candidate. Notice that in our parameter scan we
have anyway considered DM masses heavier than 100 GeV
and a sizable mass splitting between the DM and the
lightest electrically charged fermion E1. If these requirements
are dropped, one could achieve an enhanced DM
annihilation cross section at the Higgs “pole”, i.e. m N1 mh /2, or
through coannihilations (we will discuss this scenario in the
next section in the context of the 2 Higgs doublet model),
eventually relaxing the tensions with DD. This possibility
has been considered e.g. in [
56
] in a similar model (notice
that in this reference the VL neutrinos have also Majorana
mass terms and their interactions with the Z are weaker than
in the model discussed here), as the one discussed here, but
focused on the case of rather light VLLs enhancing the decay
branching fraction of the SM Higgs to diphotons (we have
instead considered the case in which this coincides with the
SM expectation).
2.5 Vacuum stability
As will be discussed in greater detail in the next section, the
presence of vector-like fermions has a very strong impact
on the behavior of the theory with respect to radiative
corrections. The RG evolution of the parameters of the scalar
potential typically suffers the strongest influence from the
introduced new physics.
As is well known, the stability of the EW vacuum depends
on the sign of the quartic coupling λ of the Higgs potential.
This parameter, positive at the electroweak scale, is driven
towards negative values, at high energy scales, by radiative
corrections mostly relying on the Yukawa coupling of the
top quark. Detailed studies, like e.g. [
57
] have shown that
the EW vacuum is stable up to energies close to the Planck
scale. A quantitative determination is nevertheless extremely
sensitive to the value of the mass of the top quark.
The presence of vector-like fermions tends to make steeper
the decrease of λ at high energy as can be seen by the
oneloop β function:
1
βλ = 16π 2 [βλ,SM + 4λ((yhNL )2 + (yhNR )2 + (yhEL )2 + (yhER )2)
− 4((yhNL )4 + (yhNR )4 + (yhEL )4 + (yhER )4)]
(26)
where βλ,SM accounts for the SM contribution.
We have then checked the stability of the EW vacuum
in the presence of the new vector-like fermions by solving
the coupled RGE’s of the Higgs quartic couplings and of
relevant parameters as the yukawa couplings of the VLF and
of the top quark and the gauge couplings.6 For simplicity we
have assumed that all the new particles lie at a same scale
m F = 400 GeV and that their Yukawa couplings are zero
below this scale. As discussed in the previous subsection,
the coupling y NL is constrained, by DM phenomenology, to
h
be very small, below 10−2. We also recall that we customarily
assume y NR = y ER = 0 to automatically comply with the
h h
constraints on the Higgs signal strength. In this simplified
picture the vacuum stability depends, besides the SM inputs,
on just one new parameter, i.e. yhEL .
The behavior of the Higgs quartic parameter with the energy
is shown, for some assignations of yhEL , in the first panel of
Fig. 3. As is evident, values of yhEL are equal to or greater
than one correspond to a fast drop of λ. For yhEL = 2 we
notice indeed that the Higgs quartic coupling becomes
negative in proximity of the energy threshold corresponding to
the mass of the VLF. Our results have been made more
quantitative in the second panel of Fig. 3. Here we have
indeed defined the stability scale UV, adopting the criterion
λ( UV) = −0.07 [
58
]. The energy scale UV is interpreted
as the scale below which new degrees of freedom should be
added in order to have stability of the EW up to high scales.
As will be further remarked along the paper, building a UV
complete model is not in the purposes of present work; as a
consequence we will implicitly adopt the minimal
requirement that UV lies above the energy scales accessible to
collider studies, namely few TeVs, so that our model provides a
reliable low energy description of the relevant
phenomenology.
3 Two Higgs doublet models
Let us now move to the case of 2HDM scenarios. We will
summarize below their most salient features and fix, as well,
6 Our analysis is rather qualitative since based on one-loop β functions.
Our results are nevertheless in good agreement with the more detailed
study presented in [
58
].
the notation. For a more extensive review we refer instead,
for example, to [
59
].
In this work we have considered the following,
CPconserving, scalar potential:
† †
V ( H1, H2) = m211 H1 H1 + m222 H2 H2
− m212( H1† H2 + h.c.) + λ21 ( H1† H1)2
+ λ22 ( H2† H2)2
† † † †
+ λ3( H1 H1)( H2 H2) + λ4( H1 H2)( H2 H1)
+ λ25 [( H1† H2)2 + h.c.],
where, as usual, v2/v1 = tan β ≡ tβ . The most general scalar
potential would feature two additional quartic couplings λ6,
λ7 which have been, for simplicity, set to zero (this can be
achieved by imposing a discrete symmetry [
60
]).
The spectrum of physical states is constituted by two
CPeven neutral states, h, identified with the 125 GeV Higgs, and
H , the CP-odd Higgs A and finally the charged Higgs H ±.
The transition from the interaction basis ( H1, H2)T to the
mass basis (h, H, A, H ±) depends on two mixing angles, α
and β. Throughout this work we will assume the so-called
alignment limit, i.e. α β − π/2. This is a reasonable
assumption since, in most scenarios, as also shown in Fig. 4,
only small deviations from the alignment limit are
experimentally allowed. In this limit, the h boson becomes
completely SM-like. A second relevant implication is that the
couplings of the second CP Higgs H with W and Z bosons
are zero at tree level, being proportional to cos(β − α)
(analogous tree-level couplings for the A boson are forbidden by
CP conservation). For a more detailed treatment of the
alignment limit, we refer the reader to e.g. Refs. [
61–65
].
The quartic couplings of the scalar potential (27) can be
expressed as a function of the masses of the physical states
as
1
λ1 = v2 [m2h + (m2H − M 2)tβ2],
1
λ2 = v2 [m2h + (m2H − M 2)tβ−2],
1
λ3 = v2 [m2h + 2m2H± − (m2H + M 2)],
1
λ4 = v2 [M 2 + m2A − 2m H± ],
1
λ5 = v2 [M 2 − m2A],
where M ≡ m12/(sβ cβ ). Unitarity and boundedness from
below of the scalar potential impose constraints on the value
of the couplings λi=1,5 [
59,66
] which, through Eq. 29, are
translated into bounds on the physical masses. In particular
these bounds imply that it is not possible to assign their values
independently one from each other. All these bounds can be
found, for example, in Refs. [
66,67
], but, for completeness,
we will report them below. For the scalar potential to be
bounded from below, the quartics must satisfy
λ1,2 > 0, λ3 > − λ1λ2, and λ3 + λ4 − |λ5| > − λ1λ2,
while s-wave tree-level unitarity imposes the requirement
|a±| , |b±| , |c±| , | f±| , e1,2 , | f1| , | p1| < 8π,
where
3
a± = 2 (λ1 + λ2) ±
1
b± = 2 (λ1 + λ2) ±
1
c± = 2 (λ1 + λ2) ±
9
4 (λ1 − λ2)2 + (2λ3 + λ4)2,
(λ1 − λ2)2 + 4λ24,
(λ1 − λ2)2 + 4λ25,
e1 = λ3 + 2λ4 − 3λ5, e2 = λ3 − λ5,
f+ = λ3 + 2λ4 + 3λ5, f− = λ3 + λ5,
f1 = λ3 + λ4, p1 = λ3 − λ4.
Vacuum stability finally requires [
68
]:
m212 m211 − m222 λ1/λ2
tan β − 4 λ1/λ2
> 0
where the mass parameters m11, m22, m12 should satisfy
m211 +
λ1v2 cos2 β
2
+
λ3v2 sin2 β
2
= tan β m212 − (λ4 + λ5)
v2 sin 2β
4
,
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
v2 sin 2β
4
.
Later on, we will include these constraints as well when doing
our scans.
The coupling of the SM fermions with the Higgses are
described by the following lagrangian:
− Lyuk = f =u,d,l mvf [ξh f f h + ξH f f H − i ξAf f γ5 f A]
SM f f
−
√2
v
u mu ξ Au PL + md ξ Ad PR d H +
√2
+ v ml ξAlνL lR H + + h.c. ,
where the parameters ξhf,H,A depend the couplings of the SM
fermions with the two doublets H1,2. Motivated by the
nonobservation of flavor-changing neutral currents (FCNCs)
we consider four different sets of ξhf,H,A corresponding to
four 2HDM models, i.e. type-I, type-II, lepton-specific and
flipped, featuring the absence of tree-level FCNCs [
59
]. The
values of the ξ for these four flavor-conserving types of
2HDMs are listed below in Table 1. On the contrary, we
assume generic couplings of the VL fermions to both H1,2
doublets7:
− LVLL = yiUR DL H˜i UR + yiUL UL H˜i†DR
+ yiDR DL Hi DR + yiDL DL Hi†DR
+ MDDL DR + MU UL UR
+ MD DL DR + h.c.,
where a sum over i = 1, 2 is implied. It is possible to define
the Yukawa couplings, yhX and yHX , by the physical CP-even
states through the following rotations:
y X
h
y X
H
HSM
HNP
cβ sβ
= sβ −cβ
cβ sβ
= sβ −cβ
y X
1
y X ,
2
H1 ,
H2
where we used the superscript X = UL/R or DL/R . As we
are working in the alignment limit, HSM becomes the SM
7 Since we are coupling the VLFs to both doublets, we cannot
rigorously refer to type-I, type-II, lepton-specific or flipped 2HDMs, as flavor
violating Yukawa couplings, possibly responsible for FCNCs, might be
induced radiatively by the VLLs. We will nevertheless retain the
classification of the various 2HDM realizations in order to distinguish the
different dependence on tan β of the couplings of the SM fermions and
the Higgs mass eigenstates.
(38)
(39)
(40)
(41)
Table 1 Couplings of the
Higgses to the SM fermions as a
function of the angles α and β
and in the alignment limit where
(β − α) → π/2
ξhu
ξhd
ξhl
ξ Hu
ξ Hd
ξHl
ξ Au
ξ Ad
ξAl
Type I
cα/sβ → 1
cα/sβ → 1
cα/sβ → 1
sα/sβ → −tβ−1
sα/sβ → −tβ−1
sα/sβ → −tβ−1
t−1
β
−tβ−1
−tβ−1
H +
Higgs double, while HNP = . Since we
(H − i A)/√2
are coupling the VL fermions to both doublets, the value of
tβ or the chosen type of 2HDM will be irrelevant for the
VLF coupling to the scalars. On the contrary, the Yukawa
couplings of the SM fermions are dictated exactly by the
choices of tβ and of the 2HDM type.
A DM candidate is again straightforwardly introduced by
considering a lagrangian of the form (40) with U ≡ N and
D ≡ E . Our analysis will substantially follow the same lines
as in the case of VLL extensions of the SM Higgs sector.
Before determining the DM observables and comparing them
with experimental constraints, we will reformulate, in the
next subsections, for the case of the 2HDM, the constraints
from the SM Higgs signal strength and from EWPT. We will
also consider an additional set of constraints, which influence
the size of the new Yukawa couplings, from the UV behavior
of the theory.
3.1 Higgs signal strengths
Having imposed the alignment limit, the extended Higgs
sector does not influence the decay branching fractions of the
125 GeV SM-like Higgs. The only possible source of
deviation from the SM expectation is represented by the VLLs,
which can affect the h → γ γ signal strength, μγ γ . The
corresponding contribution substantially coincides with the one
determined in the one Higgs doublet scenario, namely Eq. (8).
Assuming the presence of only one family of VLLs, the
simplest solution for having an experimentally viable scenario is
to set to zero one of the yhEL,R couplings. Unless stated
otherwise, we will assume, in the analysis below, that yhER = 0.
3.2 EWPT constraints
In a 2HDM+VLL framework new contributions, with
respect to the SM, to the S and T parameters originate from
both the fermionic and the scalar sector. As regards the
forType II
cα/sβ → 1
−sα/cβ → 1
−sα/cβ → 1
sα/sβ → −tβ−1
cα/cβ → tβ
cα/cβ → tβ
t−1
β
tβ
tβ
Lepton-specific
cα/sβ → 1
cα/sβ → 1
−sα/cβ → 1
sα/sβ → −tβ−1
sα/sβ → −tβ−1
cα/cβ → tβ
t−1
β
−tβ−1
tβ
Flipped
cα/sβ → 1
−sα/cβ → 1
cα/sβ → 1
sα/sβ → −tβ−1
cα/cβ → tβ
sα/sβ → −tβ−1
t−1
β
tβ
−tβ−1
mer, these contributions depend, as for the case of one Higgs
doublet, on the masses of the new fermions and their
couplings y NL,R,EL,R with the SM-like Higgs, while the
couh
plings with the other Higgs states are unconstrained. The
contributions from the scalar sector are instead related to
the masses of the new Higgs states. Also in this case it is
possible to forbid deviations from the SM expectations of
the T parameter by imposing a custodial symmetry. In the
alignment limit this is realized by setting m H m H± or
m A m H± [
46,69
] and considering only constraints from
the S parameter. As already pointed out and further clarified
below, this choice would imply excessive limitations to DM
phenomenology. For this reason we will not impose a
custodial symmetry, neither to the fermionic nor to the scalar
sector, but rather freely vary the corresponding parameters
and require in turn that the S, T parameters do not deviate
by more than 3σ from their best fit values.
For illustrative purposes we have reported in Fig. 5 the
regions allowed by EWPT for some definite assignation of
model parameters. More specifically we have fixed the
values of the DM candidate MN1 and of the lightest charged
new fermion m E1 , as well as the Yukawa coupling yhNL , to,
respectively, 120 GeV, 250 GeV and 0.01 (this very low value
is motivated by constraints from DM DD), while we have
varied the parameter yhEL , since it will be relevant for the
DM relic density as well as for LHC detection prospects.
Regarding the scalar sector we have fixed m A = 500 GeV
(left panel) and m A = 800 GeV (right panel) and varied the
mass of the CP-even Higgs state H and of the charged one
H ±. For yhEL ≤ 1 the effect of the fermionic sector on the
EWPT is subdominant such that the allowed regions
substantially corresponds to the one allowed in the case of no
VLLs present in the theory. On the contrary, once the value
of yhEL is increased, a cancellation between the contributions
from the fermionic and scalar sectors is needed in order to
comply with experimental constraints. As a consequence, the
allowed regions of the parameter space are reduced to rather
narrow bands. We also notice that, in this last case, the
conertheless assumed alignment limit). As we can see, values
of yhEL above 3 are excluded for m A = 500 GeV while for
m A = 800 GeV we get the even stronger constraint yhEL 2.
3.3 Constraints from RGE evolution
The extension of the Higgs sector with VLFs suffers also
constraints from theoretical consistency. Indeed, the presence of
new fermions affects the RGE evolution of the parameters of
the 2HDM, in particular the gauge couplings and the quartic
couplings of the scalar potential [
70
], making it difficult for
the new states to induce sizable collider signals, like diphoton
events [
71–78
] (see also below).
As regards the gauge couplings, their β functions receive a
positive contribution depending on the number of families of
vector-like fermions and on their quantum numbers under the
SM model gauge group. In case that these contributions are
too high the gauge couplings can be lead to a Landau pole
at even moderate/low energy scales. However, in the case
considered in this work, i.e. one family of vector-like leptons,
we have only a small contribution to the β functions of the
couplings g1 and g2 which does not affect in a dangerous
way their evolution with energy.
Very different is, instead, the case of the quartic couplings.
As already seen in the case of a SM-like Higgs sector, the
radiative corrections associated to the VLLs depend on their
Yukawa couplings. The β functions are given by
1
βλ1 = βλ1,2HDM + 8π 2
1
βλ2 = βλ2,2HDM + 8π 2
1
βλ3 = βλ3,2HDM + 16π 2 λ3
λ1
λ2
L
L
L
L
|y1L |2 −
|y1L |4 , (42)
|y2L |2 −
|y2L |4 , (43)
(|y1L |2 + |y2L |2)
Fig. 5 Impact of EWPT constraints in the bidimensional plane
(m H , m H± ) for two fixed assignations of m A, i.e. 500 and 800 GeV.
The blue, purple, orange and red regions represent the allowed
parameter space for, respectively, yhEL = 0.5, 1, 2, 3. The green points
represent the configurations allowed by the constraints reported in Eqs. (34)
and (35)
straints from EWPT disfavor mass degenerate H, A, H ±. We
recall that, on the other hand, the variation of the masses of
the Higgs states is constrained by perturbativity and unitarity
limits, Eqs. (34) and (35). We have then reported on Fig. 5
the regions allowed by the latter constraints, determined by
varying the input parameters of Eq. (29) over the same ranges
considered in [
66
] (contrary to this reference we have
nevL
− 2y1EL y2EL y1NL y NL
2 + (|y1NL |2 + |y1EL |2)(|y2NL |2 + |y2EL |2)
− 2y1ER y2ER y1NR y NR
2 + (|y1NR |2 + |y1ER |2)(|y2NR |2 + |y2ER |2) ,
1
βλ4 = βλ4,2HDM + 16π 2 λ4
(|y1L |2 + |y2L |2)
L
− 2y1EL y2EL y1NL y NL
2 + (|y1NL |2 − |y1EL |2)(|y2NL |2 − |y2EL |2)
+ 2y1ER y2ER y1NR y NR
2 + (|y1NR |2 − |y1ER |2)(|y2NR |2 − |y2ER |2) ,
1
βλ5 = βλ5,2HDM + 16π 2
λ4
L
(|y1L |2 + |y2L |2) − 2
|y1L |2|y2L |2 ,
L
(44)
(45)
(46)
where βλi ,2HDM are the contributions to the β function
originating only from the quartic couplings themselves and the
Yukawa couplings of the SM fermions. We refer to [
59
] for
their explicit expressions.
To simplify the notation we have expressed, in Eqs. (42)–
(46),8 the Yukawa couplings in the (H1, H2) basis.
As evident, the quartic couplings receive large radiative
corrections scaling either with the second or the fourth power
of the Yukawa couplings. As a consequence, vacuum stability
and/or perturbativity and unitarity might be spoiled at some
given energy scale unless additional degrees of freedom are
introduced in the theory.
A quantitative analysis would require the solution of Eqs.
(42)–(46) coupled with RGE for the gauge and Yukawa
couplings as function of the masses of the Higgs eigenstates and
the parameters M and tβ , which determine the initial
conditions for λ1,5, and verify conditions 34 and 35 as function
of the energy scale. A good qualitative understanding can be
nevertheless achieved by noticing that for sizable Yukawa
couplings the β functions (42)–(46) are dominated by the
negative contributions scaling with the fourth power of the
Yukawas themselves (their β function are positive, scaling
qualitatively as βy ∝ y3). As a consequence one can focus,
among 34 and 35, on the conditions λ1,2 > 0. In analogous
fashion to the case of the SM Higgs sector, discussed in the
previous section, we will just require low energy viability of
the model. In other words, a given set of model parameters
will be regarded as (at least phenomenologically) viable if the
scale at which the couplings λ1, λ2 become negative is
considerably above few TeVs, i.e. far enough from the energy
scales probed by collider processes. Additional degrees of
freedom at high energy scale might, at this point, improve
the behavior of the theory. The study of explicit scenarios is
anyway not in the purposes of this study.
According to the discussion above, in a
phenomenologically viable setup, the quartic couplings λ1 and λ2 should
not vary too fast with the energy. As proposed in [
79
], a good
approximate condition consists in imposing |βλ1,2 /λ1,2| < 1,
with λ1,5 computed according to Eq. (29) and the Yukawa
couplings set to their input value at the EW scale. In case
this condition is not fulfilled, the functions βλ1,2 would vary
too fast with the energy so that the theory would manifest
a pathological behavior already in proximity of the energy
threshold corresponding to the masses of the VLLs.9
As already pointed out, the requirements of a reliable
behavior of the theory under RG evolution mostly affect
8 Notice that even if the couplings λ6 and λ7 have been set to zero, they
are radiatively generated. So one should also consider their β function
as well as additional terms in Eqs. (42)–(46). For simplicity we have
not explicitly reported these contributions but we have included them
in our numerical computations.
9 We remark that our discussion has mostly qualitative character since
it is based on one-loop β-functions.
possible predictions of LHC signals. As will be reviewed
in greater detail in the next subsections, one of the most
characteristic signatures induced by the VLLs are enhanced
diphoton production rates from decays of resonantly
produced H/ A states. This happens because their effective
couplings with photons are increased by triangle loops of
electrically charged VLLs such that, once their masses are fixed,
the corresponding rate depends on the size of the Yukawa
couplings. The constraints from RGE can be used to put an
upper limit on the size of the Yukawa couplings which imply,
in turn, an upper limit on the diphoton production cross
sections which are expected to be observed.
As illustration we have thus reported in Fig. 6 the
isocontours of σ ( pp → A)Br ( A → γ γ ) as function of yl = y EL
h
and yL = yHEL = −yHER = −yHNL = yHNR (see below for
clarification), for two values of m A, namely 500 and 800
GeV. As further assumption we have set m E1 = m A/2 in
order to maximize the effective coupling between A and the
photons.10
Clearly, in order to obtain sensitive deviations from the
prediction of a 2HDM without VLLs, which is approximately
1 and 0.05 fb for the two examples considered, rather high
values of the new Yukawas are needed,11 which would induce
too large radiative corrections to the quartic couplings of
the scalar potential. In theoretically consistent realizations,
the VLLs have negligible effects on the diphoton production
cross section.
We have checked the validity of the criteria |βλ1,2 /λ1,2| ≤
1 by explicitly solving the RGE for some benchmark models.
We have reported two sample solutions in Fig. 7. Here we
have considered the same assignation of the model
parameters as in the lower panel of Fig. 6, and we have chosen
two assignations of the input values of the Yukawa
parameters yl,L . In the upper panel we have considered the set
(yl , yL ) = (0.5, 1), lying in the white region of the lower
panel of Fig. 6. Evidently, the couplings λ1,2 remain
positive up to an energy scale μ of the order of 106 GeV,
sufficiently high for the model point to be viable at least as a
phenomenological description.12 On the contrary, by
considering a parameter assignation lying in the blue region, RGEs
drive the couplings λ1,2 to negative values already at the
10 In the computation we have considered only a “perturbative”
enhancement. A further enhancement can be achieved through
nonperturbative effects [
80
], at the price of a rather strong fine tuning of
|m A/2 − m E1 |. We will not consider this case in the present work.
11 This requirement can be partially relaxed by introducing more than
one family of VLL.
12 We have explicitly checked the other conditions (34) and (35) as a
function of the energy and found that these are violated at a slightly
lower scale of 5 × 105 GeV. This difference is acceptable as it does
not affect the validity of our results: our goal is not to quantitatively
determine the scale at which the theory should be completed, but just
to set a qualitative criteria that applies to the theory at low energy.
The coupling of the DM to an additional Higgs doublet has a
twofold impact on dark matter phenomenology. First of all,
the extra neutral Higgs states constitute additional s-channel
mediators for DM annihilations and, only for the case of
H , t-channel mediators for scattering processes relevant for
direct detection. In addition, in the high DM mass regime,
they may represent new final states for DM annihilation
processes.
The coupling of the DM with the non-SM-like Higgs states
can be expressed, in the mass basis, in terms of the Yukawa
couplings yhX,H and of the mixing angles θXL,R :
yH N1 N1 =
yAN1 N1 = i
cos θNL sin θNR yHNL + cos θNR sin θNL yHNR ,
√2
cos θNL sin θNR yHNL − cos θNR sin θNL yHNR ,
√2
yH+ N1 E1 = cos θNL sin θER yHEL + sin θNL cos θER yHER
− cos θNR sin θEL yHNR − cos θNL sin θER yHNL .
The analysis of the DM phenomenology is structured in an
analogous way as the one performed in the previous
section. We will compute the DM annihilation cross section
and verify for which assignations of the parameters of the
model the thermally favored value, ∼ 3 × 10−26 cm3 s−1,
is achieved without conflicting with bounds from DM direct
detection. Given the dependence of the coupling between
the DM and the neutral Higgs states on the mixing angles
θ L,R the DM scattering cross section is still dominated by
N
the Z exchange processes so that the new couplings from
Eq. (47) mostly impact the determination of the DM relic
density.
As regards the DM relic density, we distinguish several
possibilities:
– m N1 ≤ m X /2, X = A, H, H ± and sizable mass
splitting between the DM and the other vector-like fermions.
In this case the situation is very similar to the case of
SM+VLLs. The most relevant DM annihilation channels
are again into fermion and gauge boson pairs. Recalling
that, in the alignment limit, there is no tree-level
coupling between the H, A states and the W, Z bosons, the
only annihilation processes substantially influenced by
the presence of the additional Higgs bosons are the ones
into SM fermions. In particular, s-channel exchange of
the CP-odd Higgs gives rise to a new s-wave contribution
so that the DM annihilation cross section can be
schematically written
m2N1 m2 1
σ v f f = 8π v2t |ξ At|2 ((4m2N1 − m2A)2 + m2A 2A) |yAN1 N1 |2
(48)
the couplings of the two Higgs doublets to SM fermions.
Given the dependence on the mass of the final state
fermions, A-exchange diagrams give a sizable
contribution mostly to the t¯t final state, when kinematically
open (an exception being a type-II/flipped 2HDM for
tan β 45, when a sizable contribution comes also from
b¯b).
As already pointed out, the strong DD limits, mostly
originating from t-channel Z exchange, impose the requirement
that the DM is essentially a pure SU (2) singlet with, as well,
a tiny hypercharged component. This implies also a
suppression of the couplings of the DM to the neutral Higgs states,
such that the DM is typically overproduced in the parameter
regions compatible with DD constraints. It is nevertheless
possible to achieve the correct relic density by profiting of
the resonant enhancement of the DM annihilation cross
section when the condition m N1 m H2,A is met. Notice that in
this case the DM annihilation cross section is also sensitive
to the total width of the H/ A state and thus sensitive to the
value of tan β. An illustration of the DM constraints in the
m N1 ≤ m A2,H regime is provided in Fig. 8. Here we have
compared, for two values of tan β (for definiteness we have
considered type-I 2HDM), the isocontours of the correct DM
relic density, for three assignations of m A = m H = m H±
(for all the three cases we set ME = ML so that the lightest
charged VLL, E1, has a mass of 500 GeV), and the DD
exclusion limit, as set by LUX. As already anticipated the only
viable regions are the ones corresponding to the s-channel
poles. We also notice that the shapes of the relic density
contours are influenced by the large (narrow) widths of the
resonances occurring for small (high) tan β.
Concerning indirect detection, possible signals might
originate with residual annihilation processes, at present
times, into f¯ f (mostly b¯b and t¯t when kinematically
accessible), W +W −, Z Z and Z h, since their corresponding
annihilation cross section have unsuppressed s-wave (i.e.
velocity independent) contributions. These annihilation processes
can be probed by searches of gamma-rays produced by
interactions of the primary products of DM annihilation. The
most stringent constraints come from searches in Dwarf
Spheroidal galaxies (DSph) [
81
]. These kinds of constraints
can probe thermally favored values of the DM annihilation
cross section only for DM masses below 100 GeV. As
evidenced in Fig. 9 they are then considerably less
competitive, with the exception of the resonance region, than the
ones from DD. An additional indirect signal might be
represented by gamma-ray lines produced in the annihilation
process N1 N1 → γ γ originate with a one-loop-induced
effective vertex between the pseudoscalar Higgs state A and
two photons [
6,83
]. In our setup this annihilation channel
is, however, rather suppressed so that it is not capable to
Evidently, the annihilation cross section depends, through
the factor ξ , on tan β and, in turn, on the realization of
probe the thermal DM region (see dashed yellow line in
Fig. 9).
– m N1 < m X /2, X = A, H, H ± and DM close in mass
with at least the lightest charged VLL. In this case the
DM relic density is not only accounted for by pair
annihilation of the DM particle N1 but also by coannihilation
processes of the type Ni E j → f¯ f , W ±h, W ± Z , i, j =
1, 213 (in most of our computations we have assumed
ME = ML and, then, the charged eigenstates are very
close in mass) which occur through s-channel exchange
of the W ± and the H ± or t-channel exchange of the VLLs
themselves. These kinds of process can easily be
dominant, providing a low enough mass splitting, with respect
to N1 N1 annihilation since their corresponding
annihilation rates depend on the couplings yhEL , y EL,R which are
H
not subject to the strong constraints from DD. We also
notice that coannihilations would be relevant in the case
that a custodial symmetry is imposed in the VLF sector.
The DM phenomenology in the presence of
coannihilations is illustrated in Fig. 10. We have reported, in the top
panel, the isocontours of the correct DM relic density in
the bidimensional plane (m N1 , m Em1−Nm1 N1 ). For simplicity we
have assumed yhEL = yHNL = −yHNR = −yHEL = yHE R and
considered two numerical values, i.e. 0.1 and 1. The
remaining non-zero coupling, yhNL , has been set to 10−3 in order
to evade constraints from DD. The masses of the new Higgs
states have been finally set to the value of 1 TeV in order
to avoid effects on the relic density for DM masses of few
hundreds GeV. As evidenced in the figure the correct relic
density is achieved, through coannihilation processes,
pro13 To a lower extent also the process N2 N2 → W +W − can be relevant.
Fig. 10 Top panel isocontours of the correct DM relic density in the
bidimensional plane (m N1 , mE1 −mN1 ) for tan β = 1, yhNR = yhER = 0,
mN1
yhNL = 5 × 10−3, and two assignations of yhEL = yHNL = −yHNR =
−yHEL = yHER , i.e. 0.1 and 1. Notice that we have set M = m H = m A =
m H± = 1 TeV. Bottom panel isocontours of the correct relic density,
assuming yhNL = yhEL , for two values of mE1 −mN1 , namely 5 and 10%,
mN1
and the corresponding excluded region by LUX, in, respectively, blue
and dark blue
vided that the relative mass splitting between the DM and the
lightest charged state is between approximately 2 and 10%.
We emphasize that we have chosen a much lower value of
y NL with respect to the limits shown, for example in Fig. 8.
h
This is because the almost full degeneracy between m N1 and
m E1 implies MN ML , in turn implying enhancement of
the angles θ L,R , which set the size of the coupling of the
N
DM with the Z , mostly responsible of the SI cross section.
This last feature is well evidenced in the bottom panel of
Fig. 10 where we have assumed the yhNL = y EL limit to
eash
ily compare relic density and direct detection. As is evident,
the latter is responsible of very strong constraints, reaching
almost yhNL ∼ 10−3 and almost excluding the regions at
viable DM relic density.
In the case that the DM relic density is mostly accounted
by coannihilation processes we do not expect ID signals since
the rate of this kind of processes is (Boltzmann) suppressed
at present times.
– m N1 > m X /2, X = A, H, H ± (no coannihilations): in
this case the situation is very different from the case of the
SM Higgs sector. Indeed, as the DM mass increases, new
annihilation channels become progressively open. We
have, first of all, when m N1 > m X /2, X = A, H, H ±,
the opening of annihilation channels of the type V X
where V = Z , W ±, X = A, H, H ±. By further
increasing the DM mass, annihilation channels into pairs of
Higgs states are finally reached. Among these new
channels the most efficient turn out to be the ones into W ± H ∓
and into H ± H ∓. Indeed these processes can occur
through t-channel exchange of the lightest charged state
E1 and the corresponding rates depend on the coupling
yH+ N1 E1 , which depends on parameters not involved in
direct detection processes and which might be of sizable
magnitude even for a SM singlet DM, provided that the
charged state E1 has a sizable SU (2) component. The
potentially rich phenomenology offered by the
annihilations into Higgs–gauge bosons and Higgs boson pairs
is the reason why we have not strictly imposed a
custodial symmetry in the scalar sector since it would have
imposed a too rigid structure to the mass spectrum.
Concerning possible indirect detection constraints, these
rely on gamma-ray signals originating with cascade decays
of the H ±, W ± [
84
]. Even though these annihilations are
sizable at present times the corresponding cross section has
nevertheless a non-negligible velocity dependent component.
Consequently, annihilations at present times have a smaller
rate than at thermal freeze-out and then ID constraints have
a marginal relevance in the region of parameter spaces
corresponding to viable DM relic density. This is shown, in one
example, in Fig. 11.
In order to explore the multi-dimensional parameter space
we have then employed a scan of the following parameters:
yhNL,R , yHNL,R ∈ [
10−3, 1
],
yhEL , yHEL,R ∈ [5 × 10−3, 3],
MN ∈ [100 GeV, 1 TeV] ,
Fig. 11 Isocontour (purple line) in the bidimensional plane m N1 , yhEL
of the correct DM relic density for m H± = 250 GeV. The orange region
is excluded by searches of gamma-ray signals in DSph. The coupling
yhNL has been set to 10−2 to evade constraints from DM direct detection
ME = ML ∈ [100 GeV, 1 TeV] ,
and we required that the model points pass the constraints
from EWPT, from perturbativity and unitarity of the scalar
quartic couplings, Eqs. (34) and (35), and from satisfying
the requirement of stability under RGEs, |βλ1,2 /λ1,2| < 1.
We have finally required that the correct DM relic density is
achieved. Similarly to the case discussed in the previous
section, we have disregarded the possibility of coannihilations
between the DM and other VLLs by further imposing a
minimal mass difference between these states. We have repeated
this scan for the different 2HDM realizations reported in
Table 1. Although the DM results are mostly insensitive to
the type of couplings of the Higgs states with SM fermions
the prospects for LHC searches, discussed in the next
subsection, will be different in the various cases.
The results of our analysis have been again reported, in
Fig. 12, in the bidimensional plane (m N1 , σSI). Similarly to
the case of the single Higgs doublet scenario, many points,
especially at lower values of the DM mass, are excluded
by LUX. Viable model configurations nevertheless exist,
already for DM masses of the order of 150 GeV. We notice
in particular the presence of points lying beyond the reach
of even next generation 1 Ton facilities like XENON1T and
LZ. This is because, for these configurations, the relic
density is achieved through the annihilations into H ± H ∓ and
H ±W ∓ final states, relying on the couplings yhE,LH,R , so that
very small values of the neutral Yukawa couplings can be
taken (as pointed above, in the presence of a single family of
VLLs, large deviations from the custodial limit are allowed
provided suitable assignments of the masses of the Higgs
states.)
3.5 Impact on LHC
In this section we will discuss the impact on LHC
phenomenology of the scenario under investigation. In the
subsections below we will provide an overview of the possible
relevant processes, which currently are (and will be probed
in the near future) by the LHC. These are distinguished in
three categories: production of Higgs states and decay into
SM fermions; production of the Higgs states and decay into
gauge bosons, especially photons; direct production of VLLs.
VLLs are directly involved only in the last two categories
of collider signals; it is nevertheless important to consider
as well limits/prospects from the first category of processes
since they put constraints on the masses of Higgs states and
on tan β which can, in turn, reduce the viable parameter space
for DM.
Among this rather broad variety of signals we will pay
particular attention to the diphoton production. It arises from
the resonant production, and subsequent decay into photon
pairs, of the neutral Higgs states. The VLL couplings
entering in this process are the Yukawa couplings yhE,LH,R . These
couplings control the annihilation cross sections into W ± H ∓
and H ± H ∓ final states, which mostly account for the DM
relic density in the high mass regime; furthermore, they are
influenced, through the S/ T parameters, by the values of
the neutral couplings yhN,LH,R , which are in turn strongly
constrained by DM phenomenology.
As a further simplification we will consider the CP-even
Higgs state A as the only candidate for a diphoton resonance.
As will be explicitly shown in the following, this condition
can be achieved by imposing a specific relation between the
VLF Yukawa couplings, so as to minimize the impact of
VLLs on the effective couplings between the CP-even state
H and photons and, at the same time, maximize their impact
on the effective Aγ γ coupling. This relation will allow one
to reduce the number of free parameters. This choice is also
motivated by the fact that the production cross section pp →
A of the CP-odd state is, at parity of masses, bigger than the
corresponding one of the CP-even state H . For the specific
case of the diphoton production, as already pointed out, a
further enhancement is achieved by a specific choice of the
masses of the charged VLLs. As a consequence, focusing on
the CP-odd Higgs A allows one to obtain conservative limits
which can be straightforwardly extended to the CP-even H .
Despite these simplifications, there is still a broad variety
of factors which influence the collider phenomenology of
a diphoton resonance. We thus summarize below the most
relevant cases, basically distinguished by the value of tan β:
– Low tan β, i.e. tan β = 1−7: The neutral Higgs states are
mostly produced through gluon fusion. Irrespective of the
type of couplings with the SM fermions (see Table 1), the
top coupling to the heavy scalars is the dominant among
the ones with SM fermions. This last coupling determines
almost entirely the production cross sections of the
processes pp → A/H . The H/ A resonances would then
dominantly decay into t¯t , or into a lighter neutral scalar
(whether kinematically allowed) and a gauge boson,14
except for the case of sizable branching fractions of decay
into charged and neutral VLLs (an important branching
fraction into the DM would be nevertheless in strong
tension with constraints from DM searches). In particular,
for tan β = 1, one can have very large, /M ∼ 5–
10%, decay width, given essentially by decays into t¯t .
The observation of t t¯ resonances would be an interesting
complementary signature of an eventual diphoton
resonance. Searches of this kind of signals have been already
performed at LHC Run I [
85,86
]. The gluon–gluon fusion
(ggF) mechanism can provide production cross sections
close to the experimental sensitivity only for tan β 1,
while for increasing values of tan β it gets rapidly
suppressed.
14 This possibility is contrived because the very strong H AZ coupling
would easily lead to very high decay widths, which would make difficult
the observation of resonances.
– Moderate tan β, i.e. tan β = 10 − 20: While gluon fusion
is still the most relevant production process, in a 2HDM
d
with enhanced ξH,A (type-II and lepton-specific), a
sizable contribution arises also from bb¯ fusion. Regardless
of the type of the 2HDM, the couplings between neutral
resonances and SM fermions are suppressed, with respect
to the previous scenario, so that they feature rather
narrow width, unless sizable contributions arise from decays
into VLLs (for tan β 5 unitarity and perturbativity
constraints favor a degenerate Higgs spectrum.). Large
cross sections for the process pp → A/H → τ τ are
expected in a 2HDM with enhanced ξ Hl,A, i.e. type-II
and flipped. Corresponding LHC searches [
87,88
] give
already strong limits, such that values of tan β above 10
are already excluded for m A,H < 500 GeV.
– High tan β, i.e. tan β 50: This regime occurs only
for the type-I and flipped 2HDM since the other cases
are essentially ruled out, for masses of the neutral
Higgses below approximately 1 TeV, by the limits from
pp → A/H → τ τ¯. Two rather different scenarios
correspond to these two types of 2HDM. In the flipped model
the A/H Higgs have enhanced couplings with b-quarks,
implying bb¯-fusion as dominant production process and,
possibly, a large decay width dominated by the bb¯ final
state. In the case of the type-I 2HDM the neutral
Higgses are “fermiophobic”, since all their couplings to the
SM fermions are suppressed by a factor 1/ tan β. Unless
the decays into VLLs are relevant, we have very narrow
widths, even H,A/m H,A ∼ 10−2, and a strong
enhancement of the decay branching fraction into photons.
In the following subsections we will provide an overview,
for the scenarios depicted above, of the possible relevant LHC
signals and the corresponding constraints/prospects of
detection. We have indeed identified some relevant subsets among
the parameter points providing the correct DM relic density
and in agreement with theoretical constraints. We have first
of all considered a set of points in the low, namely 1–5, tan β
regime (although we will mostly refer to type-I 2HDM, the
various 2HDM realizations do not substantially differ in this
regime, as already pointed out). To these we have added three
subsets, characterized by 10 ≤ tan β ≤ 40, for, respectively,
type-I, type-II and lepton-specific couplings of the 2 Higgs
doublets with the SM fermions. Two subsets at tan β = 50,
corresponding to type-I and flipped realizations, have been
finally included.
For our study we have adopted the cross sections
provided by the LHC Higgs Cross Section Working Group [
89
],
which have been produced with SusHi 1.4.1 [
90
]. More
specifically, for the 2HDM types with enhanced bottom
quark couplings to heavy scalars (type-II and flipped), we
have taken the gg/b¯b fusion cross sections calculated for
the hMSSM [
91,92
]. For the remaining two realizations,
namely type-I and lepton-specific 2HDMs, regardless of the
value of tan β, the only important production mechanism
is gg fusion, since b¯b fusion is suppressed not only by the
lower bottom quark luminosity, but also by the b¯b A/H
couplings, which scale as 1/ tan β. Therefore, as both top and
bottom quark couplings to the heavy scalars are
proportional to 1/ tan β for type-I and lepton-specific 2HDMs, it
follows that the effective gg A/H couplings have a
similar behavior. Consequently, for these two realizations, we
evaluated the gg fusion cross sections by simply taking the
hMSSM ggF cross section for tan β = 1 and rescaling it
by 1/ tan2 β.
3.5.1 A/H → f¯ f
We will start our analysis by considering the production
processes pp → f¯ f .
Their phenomenology is virtually identical to the pure
2HDM case. Indeed, being singlets under SU (3), the VLF do
not modify the gluon fusion production vertex; furthermore,
limits from DM phenomenology disfavor a sizable branching
fraction of decay of the Higgs states into VLLs. For the case
of DM this is easily understood by considering the strong
limits from DM direct detection which require very suppressed
couplings. A numerical check is provided on Fig. 13 for the
case of type-I 2DHM (the outcome would be analogous also
for the other types of 2HDM).
Here we have reported the branching ratios of decay into
DM pairs of the H and A bosons for model points,
generated through a parameter scan over the ranges illustrated in
the previous section, featuring a DM scattering cross
section below the current limit by the LUX experiment. The
figure clearly evidences typically suppressed or even
negligible values for these branching fractions. As shown in the
bottom panel of the figure, a very small fraction of points
for which Br ( A → N1 N1) > 10% is nevertheless present.
These points correspond to m A ≤ 2mt ; as a consequence
even for the low couplings imposed by DD constraints, the
“invisible” branching fraction of the CP-odd Higgs can be
comparable with the ones into SM fermions since, in absence
of the t¯t channel, the latter are similarly suppressed by the
Yukawa couplings (a further suppression is expected due
to the fact that the couplings with the SM fermions are all
proportional to 1/ tan β. This result is specific of the type-I
configuration. In other scenarios tan β enhancement of the
couplings of the A with bottom and τ fermions instead
occurs). On the contrary we see no points with Br ( A →
N1 N1) > 10% when the decay into top pairs is kinematically
accessible.
The couplings of the H/ A bosons with the heavier VL
neutrino and with the two VL electrons are, on the
contrary, not directly constrained by direct detection and in
Fig. 13 Decay branching ratios of the heavy CP-even (top panel) and
CP-odd (bottom panel) scalar into N1 N1, as function of their masses
principle could allow for sizable decay branching
fractions. However, in two of the pinpointed scenarios for
the correct DM relic density, i.e. s-channel resonances
and annihilations into heavy Higgses, these decay
processes are kinematically forbidden. Furthermore the
coannihilation scenario is as well contrived for what regards
collider prospects. We will then leave it aside for the
moment and postpone a dedicated discussion to Sect.
3.5.4.
Since the branching fractions of the Higgses decaying into
fermions depend on the masses of the final state fermions
themselves, sizable signals can be achieved only for t¯t , τ τ
and b¯b final states. The observation of the latter is
substantially precluded by huge SM backgrounds so only t¯t and τ τ
feature observational prospects. Tau pair searches can probe
type-II 2HDMs at moderate-to-high tβ 5, depending on
the value of m A, since in this case we have an enhancement
of the τ Yukwawa coupling to A, ξ Aτ = tβ . In a
complementary manner, t t¯ searches provide a discovery avenue for small
values of tβ , typically 3 [
93–96
], for any type of 2HDM.
However, looking for heavy scalars decaying into top quark
pairs is challenging from the experimental point of view,
since the interference between the signal and the SM
background can give rise to non-trivial dip-peak structures in the
t¯t invariant mass spectrum, which get smeared after binning,
thus reducing the visibility of a potential “bump” [
96,97
].
We also mention that the search for scalar resonances lighter
than 500 GeV decaying to t¯t pairs is not possible, as the t
and t¯ quark are not boosted enough, the selection cuts thus
being inefficient.
We have reported in Fig. 14 the τ τ production cross
section for the model points passing theoretical and DM
constraints, distinguishing, with different colors, the various
2HDM scenarios depicted above. As already stated, current
LHC constraints are mostly efficient in the 2HDM-II. They
can nevertheless also exclude low values of m A for other
2HDM realizations.
We have then focused, on the upper panel of Fig. 15, on
the 2HDM-II case, highlighting the dependence of the
collider limits on the value of tan β. As evident, values above
20 are excluded for m A up to 1 TeV. A similar exercise has
been performed on the lower panel of Fig. 15 for the case of
the pp → A → t¯t process, in the scenario of very low tan β.
As evident, all the points lie below current experimental
sensitivity. Only the points with tan β ∼ 1 lie close enough to
the experimental sensitivity in order to be probed in the near
future.
3.5.2 Diphoton signal
In this subsection we will investigate in more detail the
prospects for observing a diphoton signal. The corresponding
cross section can be schematically written as
σ ( pp →
→ γ γ ) = σ ( pp →
)Br (
→ γ γ ),
(50)
= H, A,
with
Fig. 15 Upper panel pp → τ¯τ cross section for type-II 2HDM. Lower
panel pp → t¯t for 2HDM type-I realizations in the low tan β regime.
In both plots the points follow a color code according to the value of
tan β. The gray regions are already experimentally excluded
Br (
2
→ γ γ ) ∝ |ASM + AH± + AVLL| ,
where ASM, AH± and AVLL represent, respectively, the loop
induced amplitudes by SM fermions, charged Higgs (only
present for the CP-even state H ) and VLLs.
The contribution associated to the VLLs can be written as
(51)
(52)
(53)
, (54)
(55)
AVLL =
i=1
2 v CE ii A1/2(τEi ),
m Ei
where we have used the definition
CE = ULE · YE · (URE )†.
The Yukawa couplings between the VLLs and the heavy
Higgs states are given by
H 1
YN = √2
0 y NL
H
yHNR 0
,
H 1
YE = √2
0 y EL
H
y ER 0
H
for the heavy CP-even scalar H and
A 1
YN = √2
0
y NR
H
−yHNL
0
0 yHEL
−yHER 0
for the CP-odd scalar A.
A general analytical expression for Eq. (52) would be
rather involved. We will, however, consider two
simplifying assumptions. First of all, in order to avoid dangerous
contributions to the decay branching fraction into photons
of the SM-like Higgs we will set, as done before, yhER = 0.
Note that, especially in the case of heavier VLLs, one can
relax this assumption, since the h → γ γ signal strength is
currently measured with only ∼ 10 − 20% accuracy;
nevertheless, for simplicity, we will take y ER = 0. Furthermore,
h
we will assume ME = ML , such that the mass matrix for the
charged VLLs simplifies to15
ME =
ME v yhE
0 ME
.
(56)
(57)
(58)
(59)
E
m E2 − m E1 = v yh ,
H −v
AVLL = 2m E1 + v yhE
Knowing that neither the sign of ME nor the one of yhE are
physical (both signs can be absorbed via a field redefinition),
we will consider only positive values for these parameters.
Thus, the eigenmass splitting reads
with ME = m E1 (m E1 + v yhE ) fixed in order to give m E1
as the lowest eigenvalue. Under these assumptions the heavy
scalar loop amplitudes can be written as
× y EL A1H/2(τE1 ) − A1H/2(τE2 )
H
+ yHER
m E1 + v yhE A1H/2(τE1 ) − m E1 + v yhE A1H/2(τE2 ) ,
m E1
m E1
A −v
AVLL = 2m E1 + v yhE
× y EL A1A/2(τE1 ) − A1A/2(τE2 )
H
− yHER
m E1 + v yhE A1A/2(τE1 ) − m E1 + v yhE A1A/2(τE2 ) .
m E1
m E1
To improve the detection potential of the heavy scalars
decayA
ing into diphotons, one should maximize the value of AVLL.
This task is completed by taking opposite signs for the
yHER , yHEL couplings. We can thus reduce the number of free
couplings by setting y ER = −yHEL ≡ yHE . In this setup the
H
15 In fact, we checked that such a texture for the mass matrix suppresses,
in a similar way as for h → γ γ , the VLL contributions to h → Z γ ,
thus leaving the latter decay SM-like.
Fig. 16 Expected diphoton cross section, as function of m A for the
model points featuring the correct DM relic density and pass constraints
from EWPT, perturbativity and unitarity. The red points refer to type-I
couplings of the Higgs doublets while the blue ones to the other type of
couplings considered in this work
H and A loop amplitudes become
H
AVLL = m E1 (2m E1 + v yhE )
−v 2 yhE yHE
A1H/2(τE1 )
m E1
+ m E1 + v yhE A1H/2(τE2 ) ,
A v yHE
AVLL = m E1
m E1
A1A/2(τE1 ) − m E1 + v yhE A1A/2(τE2 ) .
Note that, in the case where both E1,2 mass eigenstates are
much heavier than the scalar masses, i.e. τE1,2 → 0, the
CPeven and CP-odd amplitudes differ only through the loop
form factor:
A/H
AVLL
±v 2 yhE yHE
m E1 (m E1 + v yhE ) A1A//2H (0).
(60)
(61)
(62)
However, in the case where A1A/2(τE1 ) dominates over the
second term in the brackets from Eq. (61), which happens,
for example, if m E1 m A/2 and m E2 m E1 , the
CPA v , whereas
odd amplitude is indeed maximized: AVLL ∝ m E1
AVHLL ∝ m Ev1 m2 E2 .
We have reported, and confronted with the current
experimental limits [
98
], in Fig. 16 the predicted cross section
for pp → A → γ γ , for the model points providing viable
DM candidates. We have distinguished between the different
regimes described in the previous subsection, identified by
the type of interactions with the fermions and by the value
of tan β. As evident, the most promising scenarios are the
ones corresponding to low tan β and to tan β ∼ 50 for the
flipped 2HDM. These scenarios correspond, indeed, to the
configurations which maximize the production vertex of the
resonance: as already emphasized, for tan β ∼ 1 the gluon
fusion process is made efficient by the coupling with the top
quark, while for tan β ∼ 50 the production cross section is
enhanced by b-fusion. In the other type-I regimes, the cross
section quickly drops with the value of tan β.
In all the regimes considered the diphoton cross section
lies below the current experimental sensitivity; the
deviation from experimental sensitivity quickly reaches several
orders of magnitude as the value of m A increases. A
signal in diphoton events would be hardly observable, even in
future luminosity upgrades, for m A 700 GeV. The
reason of this outcome mostly lies in the fact that the size
of the Yukawa couplings of the charged VLLs are
limited from above by the requirement of consistency under
RG evolution and, only for yhEL , by EWPT. As a
consequence, no sensitive enhancement of the diphoton
production cross section, with respect to the 2HDM without VLLs,
is actually allowed. We notice, in addition, that in order
to comply with limits from DM phenomenology, the VLLs
should be typically heavier than the diphoton resonance. This
translates in a further suppression of the VLL triangle loop
contribution.
3.5.3 Other loop-induced processes
Given their quantum number assignments (and gauge
invariance), VLLs also induce, at one loop, decays of A/H into
Z γ , Z Z , W W , which can be probed at the LHC.
Among these processes, the cleanest signal is likely
provided by the Z γ channel. It is searched for in events with one
photon and dijets or dileptons originating from the decay of
the Z . Although the corresponding production rate is
suppressed with respect to diphoton signals, the potential signal
is particularly clean (i.e. low background), especially in the
case of lepton final states. In the setup under investigation,
the A → Z γ decay width, to a very good approximation,
reads [
45,49
]
αg2m3A
( A → Z γ ) = 512π 4v2c2
W
m Z
1 − m A
3
× At
AZγ
AZγ
+ Ab
AZγ 2
+ AVLL
.
The top-loop and bottom-loop amplitudes have simple
expressions,
AtA,bZγ = Nc Qt,b Vt,b ξ At,b A1A/2(τt,b, λt,b),
with Q f the electric charge of the SM fermion f , V f its
vectorial coupling to the Z boson, and ξ At,b defined in Table 1. For
the A1A/2(τi , λi ) loop form factors, we use the same
expressions as in Ref. [
45
], with τi ≡ mm2A2 and λi ≡ m2
m2Z .
i i
(64)
Concerning the VLL A → Z γ loop amplitude, its general
expression, which is again given in the appendix of Ref. [
45
]
(denoted as A˜ Zf γ there), is rather contrived, and will not be
displayed here. However, for our particular choice of the
charged VLL mass and pseudoscalar Yukawa matrices, it
takes the simple form
AZγ v yHE
AVLL = Qe Ve m E1
×
m E1
A1A/2(τE1 , λE1 ) − m E1 + v yhE A1A/2(τE2 , λE2 ) ,
(65)
with Qe = −1 the electric charge of the VL electron and
Ve = −0.25 + sW2 the vectorial coupling to the Z of the SM
electron. One can see that, contrary to the general case, the
diagrams with off-diagonal A and Z couplings to the VLFs
vanish for our choice of parameters. Unfortunately, due to
the smallness of Ve 0.02, our scenario does not produce
a sizable modification to the A → Z γ decay channel with
respect to the case of an ordinary 2HDM.
We also briefly comment on the case of the W W and
Z Z decay channels. As the A → γ γ /Z γ processes, both
A → Z Z and A → W W are loop-suppressed ( AW W / A Z Z
vertices are forbidden at tree level by CP-invariance).
Moreover, detection of such decays is challenging due to either (i)
suppression by reduced branching ratios (Br(Z → + −)
7%, = e, μ) or (ii) final states that are difficult to
reconstruct/disentangle from the background (hadronic decays of
W, Z and leptonic decays of the W , W → ν , which involve
missing transverse energy). Therefore, we will not consider
these channels as they are not as clean and/or competitive as
the ones already discussed.
3.5.4 Direct production of VLLs
We conclude our overview of the collider phenomenology
of the scenario under investigation by briefly commenting
on possible direct searches of the VLLs. VLLs can be
produced at LHC through the Drell–Yann processes [
27,99–
101
] pp → Z ∗/γ ∗ → E E , pp → Z ∗ → N N , and
pp → W ∗ → N E .16 The results of corresponding LHC
searches [
105,106
] cannot be, nevertheless, applied to our
case since they rely on the presence of a mixing with SM
leptons. In our scenario, in order to guarantee the stability
of the DM candidate, we have forbidden such a mixing by
imposing a Z2 symmetry under which the VLL sector is odd
and the SM is even (see next section). On the contrary, a
possible collider signal would be represented by the
production of E1 E1 or N2 E1 and their subsequent decay into
16 Alternatively VLLs might originate with the decay of produced
heavy neutral Higgses [
102–104
].
DM, which can be tested in 2–3 charged leptons plus
missing energy final state events. Searches of this kind have been
performed in the context of supersymmetric scenarios [
107–
109
]. In order to take into account possible constraints, we
have imposed (ad exception of the coannihilation regime), in
our scans, a lower limit on the mass of the lightest charged
VLL of 300 GeV. Direct production of DM, through off-shell
Z/h boson or on-shell heavy Higgses, can be instead hardly
tested, through monojet searches, since constraints from DM
direct detection imposes, in most of the phenomenologically
viable parameter space (see discussion in the previous
section) a negligible branching fraction of decay into DM pairs
(see Fig. 13).
Another potentially interesting channel would be the
production of a charged Higgs and its subsequent decay into
N1 E1, followed by E1 → N1W . However, for most of the
points providing the correct DM relic density and, at the
same time, passing the DD constraints, we have m H± <
m N1 + m E1 , so that production can occur only through
offshell charged Higgs. Furthermore, the dominant production
modes of H ± at the LHC, gg → t b H ± and gb → t H ±, are
phase-space suppressed by the top quark produced in
association and typically have a low cross section. The s-channel
production of a charged Higgs, qq → H ± is not a valid
option neither: even if the charged Higgs would be on-shell,
the low Yukawa couplings of the initial state quarks renders
such a process unobservable. For a more detailed discussion,
we refer the reader to Ref. [
27
].
We close the section by commenting again on possible
production of VLLs from decays of the neutral Higgses.
As pointed out in Sect. 3.5.1, in the coannihilation
scenario sizable branching fractions for the decays H/ A →
Ei Ei , i = 1, 2 are not forbidden by limits from DM
phenomenology. However, while E2, having a sizable
admixture of a SU (2)L doublet, almost always decays promptly
into E1 plus a W /Z / h boson (on or off-shell), the E1 can
decay only into a N1 and two fermions through an off-shell
W . This decay rate would be doubly suppressed by the very
small coupling to the W of the mostly SU(2) singlet DM
and by the phase space. Consequently, the E1 state would be
long-lived or even stable on collider scales.
3.6 Constraints on the charged Higgs
Collider limits on the charged Higgs are mostly relevant for
very light masses, namely m H± < mt . In this case, light
charged Higgses can be searched for in the t → H ±b decays,
followed by H ± → cs or H ± → τ ντ . Searches for this
processes have been performed both by ATLAS [
110
] and
CMS [
111,112
]. No sensitive variations in the top branching
fractions with respect to the SM have been detected,
disfavoring masses of the charged Higgs below 160 GeV. The ATLAS
collaboration has performed searches for H ± → τ ντ [113]
also in the high mass regime, i.e. m H± > mt , with the
charged Higgs being produced in association with a top
quark, i.e. through the process gb → t H ±. The limits
obtained, however, cannot yet constrain efficiently most of
the 2HDM setups considered in this work (with the possible
exception of the lepton specific 2HDM), since the τ ντ final
state has a low branching fraction at high masses [
114
].
The mass of the charged Higgs can also be strongly
constrained by low energy observables. As these bounds are
determined by the value of tan β, they are actually dependent
on the type of 2HDM realization. For an extensive review we
refer, for example, to Ref. [
114
]. We will instead summarize,
in the following, the constraints relevant to our analysis.
We have first of all to consider loop-induced
contributions to the B → Xs γ process. These depend on the
coupling of the charged Higgs to t ,b and s quarks. In the
typeI and lepton-specific models, all the relevant couplings are
suppressed by 1/ tan β and, hence, sizable constraints are
obtained only for very low tan β [
115
]. Much stronger bounds
are instead obtained in 2HDM-II, excluding masses of the
charged Higgs up to order of 400 GeV [
116–119
], practically
independent from the value of tan β. A second relevant bound
comes from the semileptonic decays of the pseudoscalar
mesons, in particular B(B → τ ν). By requiring the ratio
r = B(B → τ ντ )/B(B → τ ντ )SM to be consistent with the
experimental determination r = 1.56 ± 0.47 [
120,121
], one
obtains, only for the type-II 2HDM, a limit on the
bidimensional (m H± , tan β) plane which is relevant for tan β 20.
As already stated, our analysis has been mostly carried on
a purely phenomenological basis. In this subsection we will
nevertheless take some steps towards a more complete
construction discussing some potential challenges in the model
building, the stability of the DM and the suppression of
FCNCs.
VLFs with the same quantum numbers of the SM fermions
allow for yukawa coupling of one VLF, one SM fermion and
one Higgs boson, responsible for a mass mixing which makes
all the VLFs to decay into a SM fermion and a gauge or Higgs
boson. In order that the lightest neutral VLF is a viable DM
candidate this kind of mixing should be strongly suppressed
or possibly forbidden. The simplest option to achieve this
goal would be represented by the introduction of a Z2
symmetry, which we label ZVLL, under which the VLLs are odd
2
(with the SM states being instead even), so that the coupling
originating with the mixing between the VLLs and the SM
fermions would be actually forbidden.
Another potential challenge is represented by the
presence of FCNCs. FCNCs induced by the coupling of the SM
fermions with the Higgs doublets have been forbidden by
assuming four specific configurations for these couplings.
These can be realized by assuming suitable discrete
symmetries. The type-I configuration realized by assuming a
discrete symmetry Z2HDM such that H1 → −H1 so that all
2
the SM fermions are coupled to the H2 doublet, while in
the type-II configuration, also the right-handed d-quarks and
right-handed leptons are odd under this Z2 symmetry so that
they are coupled with the H1 doublet. The lepton-specific and
flipped configurations are similarly realized through suitable
assignation of the Z2 charges for up-quarks, down-quarks
and leptons. Possible UV completions for these 2HDM
realizations have been studied e.g. [
122–124
].
The addition, to the mass spectrum, of VLFs, freely
coupled to both Higgs doublets, provides a further potential
source of FCNCs, induced at one-loop in this case. The
determination of possible bounds for generic couplings of
the VLFs, as considered here, is not in the purpose of this
work. A possible simple solution would be represented by
making also some of the VLFs odd under the Z2HDM
sym2
metry. Drawing inspiration from the simple flavor conserving
2HDMs, where the SM quark doublet and the up-type
righthanded quarks are, by convention, even under Z2HDM, we
2
choose the VLL doublet and the NL,R singlet VL neutrino
(“up-type” VLL) to be, similarly, even under Z2HDM. This
2
leaves us with two possibilities for EL,R :
1. EL,R is also even under Z22HDM, meaning that all VLLs
couple to H2;
2. EL,R is odd under Z2HDM, which implies that VL
elec2
trons couple to H1, while VL neutrinos couple to H2.
As evident from the discussion above, Z2HDM and ZVLL
2 2
should be distinct symmetry groups since the VLLs have the
same charge under ZVLL but different charges under Z2HDM.
2 2
Once the two symmetries are imposed, the Higgs+VLL
Lagrangian reads as follows (for simplicity, mass terms have
been omitted since not relevant for the discussion):
†
−LV L L = yNR L L H˜2 NR + yNL N L H˜2 L R
+ yER L L Hi E R + yEL E L Hi† L R + h.c.,
where i = 1, 2 corresponds to the two cases mentioned
above. After EWSB, the interactions lagrangian of the VL
neutrinos with the neutral Higgs scalars is the same in the
two cases and reads
2 Lφ N N =
+ H
+ h.c.
0
−yNL cβ
NL† NL†
−yNR cβ
0
h
+ A
0
yNL sβ
yNR sβ
0
0
i yNL cβ
−i yNR cβ
0
E R
E R
(68)
E R
E R
(69)
(70)
(71)
(72)
on the contrary, in the case of the VL electrons we distinguish
the following two possibilities:
+ H
+ h.c.,
+ H
+ h.c..
−
√
2 L(φ1E) E =
E L† EL†
h
0
yEL sβ
yER sβ
0
+ A
0
0
yEL cβ
i yEL sβ
yER cβ
0
−i yER sβ
0
0
−i yEL cβ
i yER cβ
0
h
+ A
0
yEL sβ
yER sβ
0
As regards the interactions of the VLLs with the charged
Higgs we have
− L(H1)± N E = H + NL† NL†
0
yNL cβ
yER sβ
0
E R
E R
+ H − E L† EL†
+ h.c.,
+ H − E L† EL†
+ h.c.
− L(H2)± N E = H + NL† NL†
0
yEL sβ
yNR cβ
0
NR
NR
0
−yEL cβ
0
yNL cβ
yNR cβ
0
−yER cβ
0
NR
NR
E R
E R
(66)
NR
NR ;
(67)
The couplings introduced in this subsection can be related
to the ones used in our analysis by reabsorbing a factor sβ
(cβ ), in the case that EL,R (EL,R ) couples to H1(H2), into
the definitions of the VLL Yukawa couplings to the 125 GeV
Higgs boson, h. For the VL neutrinos, the redefined couplings
to the scalars would read
yhNL,R ≡ yNL,R sβ , yHNL,R ≡ −yNL,R cβ = −yhNL,R tβ−1,
whereas for the VL electrons we have
yhEL,R ≡ yEL,R cβ , yHEL,R ≡ yEL,R sβ = yhEL,R tβ (73)
yhEL,R ≡ yEL,R sβ , yHEL,R ≡ −yEL,R cβ = −yhEL,R tβ−1. (74)
in the cases of, respectively, couplings with H1 and H2.
We then notice that, contrary to the case where the VLLs
couple to both scalar doublets, tβ plays now a role also in
the VLL sector. More specifically, one finds, in the VLL
couplings to H, A (relative to their couplings to h),the same
type of tβ−1 suppression or tβ enhancement as for the SM
fermions.
The impact of this feature on DM phenomenology has
been sketched in Fig. 17. Here we have reported in the two
panels the isocontours of the correct DM relic density, as well
as the excluded region by LUX, in the bidimensional plane
(m N1 , yhNL ) (we have used the relations above to adopt the
same variables as the rest of the text. We have also assumed
yhNR = 0), and for three values of tan β, namely 1,10 and
20, while keeping fixed yhEL = 0.1, yhER = 0. For
numerical convenience we have fixed ML = ME = 2MN rather
than to a constant value, as the analogous plots in Sect. 2.4.
This implies a different morphology for the LUX excluded
region, with respect to the one shown in Fig. 8. This is because
keeping ML and ME fixed to a constant value, rather than
considering a constant ratio with MN , changes the behavior
of the angles θ L,R with yhNL (see Eq. 14). The choice of the
N
constant ratio implies, in particular, stronger bounds at low
DM masses.
The more constrained, with respect to general case
discussed in the rest of the text, structure of the couplings
influences the scenarios for the correct relic density, i.e. s-channel
resonances and annihilation into heavy Higgs bosons in the
following way. The relation between the couplings yhNL , yHNL
tends to disfavor the case of resonant annihilations since they
make the constraints from DD stronger with respect to the
case in which these two couplings can be regarded as
independent. The effectiveness of these constraints increases with
tan β since the couplings of the DM with the H and A bosons
are now suppressed as 1/ tan β. The regime of annihilations
into heavy Higgs bosons is perfectly viable in the case from
Eq. (73) (“Scenario I”) where the coupling y EL can be even
H
enhanced at high tan β (in particular for tan β = 20 this
annihilation is so strong that the DM results are underabundant
in the range of the parameters reported in the plot). More
contrived is instead the case from Eq. (74) (“Scenario II”)
where DM annihilation into H + H − is suppressed at high
tan β, thus increasing the tension with DD constraints. The
light DM regime is, instead, negligibly affected since the relic
density is mostly determined by the couplings of the DM with
the W and Z boson which depend only on yhNL and yhEL . For
the same reason, the constraints from direct detection do not
change by varying tan β.
3.8 Summary of results
The results of our study are summarized in Fig. 18. Here we
have put together all the results for DM phenomenology with
theoretical constraints, i.e. scalar quartic couplings RGEs,
EWPT constraints, limits from collider searches, mostly
H/ A → τ τ , and constraints from low energy observables
(for the latter we have adopted the limits on (m H± , tan β) as
reported in Refs. [
115,125
]).
The three panels of Fig. 18 show, for three regimes of
values of tan β, i.e. low, moderate and heavy (see Sect. 2.5),
in the (m N1 , σNSI1 p) plane, the model points providing the
correct DM relic density and satisfying the constraints listed
above.
The results reported in Fig. 18 can be explained as
follows. The distributions of the points in the three panels of the
figure appear to be rather similar. As discussed in the text,
under the assumption that the VLL can couple with both
Higgs doublets, the dependence on tan β of the couplings
of the DM is reabsorbed in the definition of the couplings
themselves. We notice nevertheless that light DM masses,
1. × 10−43
]
2m1. × 10−45
p cIS
[
1. × 10−47 Xenon 1T
Fig. 18 Summary plots including all the constraints discussed
throughout this work. Each of the three panels of the figure refers to a
different regime of values of tan β (see main text for details), indicated
on the top of each panel
i.e. lighter than approximately 400 GeV become
progressively disfavored as the value of tan β increases. DD limits
are mostly evaded if the DM relic density is achieved either
in correspondence of s-channel resonances or by
annihilations involving heavy Higgs bosons, in particular the charged
ones, as final states. The former possibility becomes
increasingly contrived at higher values of tan β because the reduced
decay width of the H/ A states requires a stronger fine
tuning in the |m N1 − m A,H /2| ratio (a possible exception would
be represented by the flipped 2HDM at very high, i.e. 45,
tan β). This problem is partially overcome by considering
high enough values of the masses of H and A. The case
of the annihilations into heavy Higgs bosons is influenced
by several aspects, according the configuration, i.e. type-I,
type-II, lepton-specific or flipped, chosen for the couplings
with SM fermions. The type-II configuration is excluded
for m A below 400 GeV in the moderate tan β regime, and
for considerably higher masses in the high tan β regime, by
LHC searches in the τ τ channel (cf. Fig. 14). Values of m A
below 400 GeV are also excluded in the flipped configuration
for high tan β. These constraints also partially influence the
other Higgs masses since for moderate/high tan β the
constraints 34 and 35 and EWPT tend to favor a mass degenerate
heavy Higgs spectrum. In the type-II model the mass of the
charged Higgs is, nevertheless, individually constrained to
be above approximately 400 GeV by constraints from low
energy processes.
4 Conclusions
In this work, we have performed an extensive study of the
impact of the addition of a family of vector-like fermions,
with suitable quantum numbers such as to provide a DM
candidate, to the SM and to various types of 2HDMs.
The SM+VLLs realization is strongly constrained. The
correct relic density implies too strong interactions with the
Z -boson, ruled out by DM direct detection unless the DM,
and hence in turn the whole spectrum of the new fermions,
lies above the TeV scale.
Lower DM masses can instead be achieved in 2HDM
realizations. Indeed, s-channel enhancement, in
correspondence with the H/ A poles, can provide the correct relic
density even for a small hypercharge/SU(2) component of
the DM. In addition, efficient DM annihilations can also
be achieved, in the H ± H ∓ and W ± H ∓ final states. The
corresponding cross section is not directly correlated with
the DM DD cross section, such that it would be possible
to evade current and even next future bounds. On the other
hand, the DM relic density depends on the masses of the
new Higgs states. Complementary constraints thus come
from their experimental searches. Given their dependence
on tan β the allowed parameter space actually depends on
the type of couplings of the Higgs doublets with the SM
fermions.
Type-II, and to lesser extent, flipped 2HDMs, are the most
constrained since low values of m H± (and in turn DM masses)
are excluded by low energy observables. Moreover a large
part of the type-II parameter space is excluded by limits from
searches of A/H → τ τ . Combining these constraints, DM
masses below 400 GeV are strongly disfavored. For the other
two 2HDM realizations, constraints from searches of extra
Higgses are not yet competitive with DM constraints and
lower DM masses are accessible.
Although the size of the Yukawa couplings of the charged
VLLs can account for the correct DM relic density, it does not
account for a significant enhancement of the diphoton
production rates observable at colliders. This happens because
the limits from EWPT and RGE forbid values greater than
∼ 1 for thes couplings.
Moreover, the possibility of a direct observation of the
VLLs appears similarly contrived in particular for what
regards the DM. It could be indeed produced, in a 2HDM
setup, by the decays of the neutral H/A or of the charged
H ± (in this case rather than pair DM production one would
have production of one DM state in association with the
lightest VL lepton E1) or, alternatively through the
production of virtual h/Z bosons. Concerning the DM
candidate N1, sizable couplings with the neutral Higgs bosons
(as well as with the Z boson) are disfavored especially by
DM direct detection constraints so that the corresponding
decay branching fractions are suppressed (an exception of
a small region parameter space for type-I 2HDM). In
addition, in correspondence with two of the principally
considered scenarios for the correct DM relic density, decays of
neutral bosons into DM pairs are phase-space suppressed
(e.g. in the s-channel resonance scenario) or even
forbidden. The coupling between the DM, the lightest electrically
charged VLL, and the charged Higgs is not required to be
suppressed; however, the production of N1 E1 from decays
of the charged Higgs is similarly disfavored since in most of
the viable parameter space this decay is kinematically
forbidden.
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Page 28 of 30
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